Computational Game Theory

a.Consider
a first price auction where each participant pays $1 to participate in the
auction.

Are there deterministic Nash Equilibriums?

b.Consider
a second price auction where the participant that bids the second price
receives $1 (if there are more then one, they share it).

What are the deterministic Nash
Equilibriums?

(Assume that
all the valuations and the bids are positive integer numbers.)

2.Job
Scheduling: Show how to
compute efficiently a Nash equilibrium for n
jobs and m (different speed) machines.

(Hint:
consider the jobs in a sorted order.)

3.Coordination
Ratio & Job scheduling:
Assume that all the jobs have identical weights and all the machines have
identical speeds. Show that for any fixed number of machines, m, when
the number of jobs grows, the coordination ratio is 1+o(1).

4.Selfish Routing:

a.Show
that if the delay function is l(x) = xd , for d>1,then OPT=NASH.

b.Consider
a quadratic delay function is l(x) = ax2 + bx +1, for some
constants a andb. Derive an example
that NASH differs from OPT (as much as you can).